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\begin{document}
\title{Homework 3}
\author{Introduction to Robotics\\
ncorrell@colorado.edu}
\date{} %%If commented, the current date is used.
\maketitle


\begin{enumerate}
\item An ultrasound sensor measures distance $x=c\Delta t/2$. Here, $c$ is the speed of sound and $\Delta t$ is the difference in time between emitting and receiving a signal.
Let the variance of your time measurement $\Delta t$ be $\sigma_t^2$. What can you say about $x$, when $c$ is assumed to be constant? Hint: how does a change in $\Delta t$ affect $x$? \item Consider a unicycle that turns with angular velocity $\dot{\phi}$ and has radius $r$. Its speed is thus a function of $\dot{\phi}$ and $r$ and is given by
\begin{equation}
\nonumber
v=f(\dot{\phi},r)=r\dot{\phi}
\end{equation}
Assume that your measurement of $\dot{\phi}$ is noisy and has a standard deviation $\sigma_{\phi}$.  Use the error propagation law to calculate the resulting variance of your speed estimate $\sigma_v^2$.
\item Assume that the ceiling is equipped with infra-red markers that the robot can identify with some certainty. Your task is to develop a probabilistic localization scheme, and you would like to calculate the probability $p(marker|reading)$ to be close to a certain marker given a certain sensing reading and information about how the robot has moved.
\begin{enumerate}
\item Derive an expression for $p(marker|reading)$ assuming that you have an estimate of the probability to correctly identify a marker $p(reading|marker)$ and the probability $p(marker)$ of being underneath a specific marker.
\item Now assume that the likelihood that you are reading a marker correctly is 90\%, that you get a wrong reading is 10\%, and that you do not see a marker when passing right underneath it is 20\%. Consider a narrow corridor that is equipped with 4 markers. You know with certainty that you started from the entry closests to marker 1 and move right in a straight line. The first reading you get is `` Marker 3''. Calculate the probability to be indeed underneath marker 3.
\item Could the robot also possibly be underneath marker 4? \end{enumerate}
\end{enumerate}

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